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Statement

Theorem 2 (p. 9). Let $A_1,A_2,\ldots$ be a finite or infinite sequence of finite sets satisfying

$$|A_i|\ge2\quad\text{and}\quad\prod_i\Bigl(1-\frac1{2^{\alpha_i}}\Bigr)\ge\frac12,$$

where $\alpha_i=|A_i|$, and let $A_1',A_2',\ldots$ be a finite or infinite sequence of infinite sets. Then the family $\{A_i\}\cup\{A_i'\}$ has property B: some set meets every member of the family and contains none.

Proof pointer

None in the paper. It introduces the theorem (p. 9) as provable by slightly more complicated arguments than those for Theorem 1 and gives no proof.

Read depth

Claims checked: the statement was read clause by clause on the page image of the print. The paper gives no proof, so none was checked. Nothing here is independently reviewed.

Dependencies

Its case of finitely many $A_i$ and no $A_i'$ is case (4) of Theorem 1.

Source. P. Erdős, On a combinatorial problem, Nordisk Mat. Tidskr. 11 (1963), 5--10, 40; the edition read is named on the source card.

Bears on

• Problem 901: the problem concerns finite uniform families, which Theorem 1 already covers; this extension to infinite families adds nothing to the bounds on $m(n)$.